a b s t r a c tA binary three-dimensional (3D) image I is well-composed if the boundary surface of its continuous analog is a 2D manifold. In this paper, we present a method to locally ''repair'' the cubical complex Q (I) (embedded in R 3 ) associated to I to obtain a polyhedral complex P(I) homotopy equivalent to Q (I) such that the boundary surface of P(I) is a 2D manifold (and, hence, P(I) is a well-composed polyhedral complex). For this aim, we develop a new codification system for a complex K , called ExtendedCubeMap (ECM) representation of K , that codifies: (1) the information of the cells of K (including geometric information), under the form of a 3D grayscale image g P ; and (2) the boundary face relations between the cells of K , under the form of a set B P of structuring elements that can be stored as indexes in a look-up table. We describe a procedure to locally modify the ECM representation E Q of the cubical complex Q (I) to obtain an ECM representation of a well-composed polyhedral complex P(I) that is homotopy equivalent to Q (I). The construction of the polyhedral complex P(I) is accomplished for proving the results though it is not necessary to be done in practice, since the image g P (obtained by the repairing process on E Q ) together with the set B P codify all the geometric and topological information of P(I).
Starting from an nD geometrical object, a cellular subdivision of such an object provides an algebraic counterpart from which homology information can be computed. In this paper, we develop a process to drastically reduce the amount of data that represent the original object, with the purpose of a subsequent homology computation. The technique applied is based on the construction of a sequence of elementary chain homotopies (integral operators) which algebraically connect the initial object with a simplified one with the same homological information than the former.
a b s t r a c tThis paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AMmodel. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here, extending the work done in [R. González-Díaz, P. Real, On the cohomology of 3D digital images, Discrete Appl. Math. 147 (2005) 245-263] in which the ground ring was a field. The concept of generators which are ''nicely'' representative is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse).
Abstract. This paper introduces an algebraic framework for a topological analysis of time-varying 2D digital binary-valued images, each of them defined as 2D arrays of pixels. Our answer is based on an algebraictopological coding, called AT-model, for a nD (n = 2, 3) digital binaryvalued image I consisting simply in taking I together with an algebraic object depending on it. Considering AT-models for all the 2D digital images in a time sequence, it is possible to get an AT-model for the 3D digital image consisting in concatenating the successive 2D digital images in the sequence. If the frames are represented in a quadtree format, a similar positive result can be derived.
In this paper, we deal with the problem of the computation of the homology of a finite simplicial complex after an "elementary simplicial perturbation" process such as the inclusion or elimination of a maximal simplex or an edge contraction. To this aim we compute an algebraic topological model that is a special chain homotopy equivalence connecting the simplicial complex with its homology (working with a field as the ground ring).
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